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MTX 9100 Lecture 6 size effects

Size effects
Lecture 6
-Why does size influence the material’s properties?
-How does size influence the material’s performance?
-Why are properties of nanoscale objects different
than those of the same materials at the bulk scale?
-Why nanomaterials are unstable?
Size-dependent properties
At the nanometer scale, properties become size-dependent.
For example,
(1) Chemical properties – reactivity, catalysis
(2) Thermal properties – melting temperature
(3) Mechanical properties – adhesion, capillary forces
(4) Optical properties – absorption and scattering of light
(5) Electrical properties – tunneling current
(6) Magnetic properties – superparamagnetic effect
New properties enable new applications
Materials structures
Most materials are made up of ordered crystals
that meet at disordered boundaries; the crystals in
nanomaterials are only 100–10,000 atoms across.
Amorphous or “glassy” materials are totally
disordered; the only characteristic dimension is
that of the atoms or molecules that make them up.
They are an extreme from of nanomaterial.
Thermal property - Melting point
Thermal property - Melting temperature
Melting Point (Microscopic Definition)
Temperature at which the atoms, ions, or
molecules in a substance have enough
energy to overcome the intermolecular
forces that hold the them in a “fixed”
position in a solid
At macroscopic length scales,
the melting temperature of
materials is size-independent.
For example, an ice cube and a
glacier both melt at the same
Thermal properties
Nanocrystal size
surface energy
melting point
In contact with
3 atoms
In contact with
7 atoms
Surface atoms require less
energy to move because they
are in contact with fewer
atoms of the substance
3 nm CdSe nanocrystal melts at 700 K compared to
bulk CdSe at 1678 K
Melting point as a function of size
Thermal transport
Heat is transported in materials by two different
lattice vibration waves (phonons) and
Free electrons.
In metals, the electron mechanism of heat transport is
significantly more efficient than phonon processes.
In the case of nonmetals, phonons are the main mechanism of
thermal transport.
In both metals and nonmetals, as the system length scale is
reduced to the nanoscale, there are quantum
confinement and classical scattering effects.
Quantum confinement
The presence of nearby surfaces in 0-D, 1-D, and 2-D
nanostructures causes a change in the distribution of the
phonon frequencies as a function of phonon wavelength as
well as the appearance of surface phonon modes.
These processes lead to changes in the velocity with which
the variations in the shape of the wave’s amplitude propagate,
the so-called group velocity.
The phonon lifetime is modified due to phonon-phonon
interaction and free surface and grain boundary scattering.
Thermal property - Conductivity
where v is a particle velocity, l is a free path
length, С = сn is a heat capacity of unit
volume, c is a heat capacity of single particle,
n is a number of particles
Mechanical Properties
At the nanoscale, surface and interface
forces become dominant.
For example,
(1) Adhesion forces
(2) Capillary forces
(3) Strain forces
These forces can exceed
forces that are normally
dominant at macroscopic
length scales
Mechanical properties
Relative to microstructural (MSM) metals and alloys, the NSM
contain a higher fraction of grain boundary volume (for example,
for a grain size of 10 nm, between 14 and 27% of all atoms reside
in a region within 0.5–1.0 nm of a grain boundary);
grain boundaries play a significant role in the materials
Changes in the grain size result in a high density of incoherent
interfaces or other lattice defects such as dislocations,
vacancies, etc.
As the grain size d of the solid decreases, the proportion of
atoms located at or near grain boundaries relative to those within
the interior of a crystalline grain, scales as 1/d.
This has important implications for properties in ultra-finegrained materials which will be principally controlled by interfacial
properties rather than those of the bulk.
Grain boundaries
Crystals contain internal interfacial defects, know as
grain boundaries, where the lattice orientation changes
The misfit between adjacent crystallites in the
grain boundaries changes the atomic structure
(e.g. the average atomic density, the nearestneighbor coordination, etc.) of materials.
At high defect densities the volume fraction of
defects becomes comparable with the volume
fraction of the crystalline regions.
In fact, this is the case if the crystal
diameter becomes comparable with the thickness
of the interfaces.
Non – equilibrium materials
Crystals always contain defects
Vacancies are point defects in the
crystalline structure of a solid that may
control many physical properties in
materials such as conductivity and
However, nanocrystals are predicted to be
essentially vacancy-free; their small size
precludes any significant vacancy
This result has important consequences for
all thermo mechanical properties and
processes (such as creep and precipitation)
which are based on the presence and
migration of vacancies in the lattice.
Point defects: 0.1 nm (10-10 m)
Impurity atoms
Material properties can be altered
significantly through the addition
of impurity atoms
Point defects - Imperfections, such as vacancies, that are
located typically at one (in some cases a few) sites in the
Extended defects - Defects that involve several atoms/ions
and thus occur over a finite volume of the crystalline material
(e.g., dislocations, stacking faults, etc.).
Vacancy - An atom or an ion missing from its regular
crystallographic site.
Interstitial defect - A point defect produced when an atom is
placed into the crystal at a site that is normally not a lattice
Substitutional defect - A point defect produced when an atom
is removed from a regular lattice point and replaced with a
different atom, usually of a different size.
Summary of point defects
(c) 2003 Brooks/Cole Publishing / Thomson Learning
(a) vacancy, (b) interstitial atom, (c) small substitutional atom, (d) large
substitutional atom, (e) Frenkel defect, (f) Schottky defect.
Defects for plasticity
Crystals all contain line defects known as
Dislocations act
the main
source of
deformation in
Plastic deformation
(a) When a shear stress is applied to the dislocation in (a),
the atoms are displaced, causing the dislocation to move
one Burgers vector in the slip direction (b). Continued
movement of the dislocation eventually creates a step (c),
and the crystal is deformed. (Adapted from A.G. Guy, Essentials of
Materials Science, McGraw-Hill, 1976.) (d) Motion of caterpillar is
analogous to the motion of a dislocation.
Dislocations are
positioned closer
together and dislocations
movement in the net is
hindered by interaction
between them. Together
with the reduced elastic
strain energy, this fact
results in dislocations
that are relatively
immobile and
the imposed stress
necessary to deform a
material increases with
decrease in grain size.
Dislocations have a less dominant role to play
in the description of the properties of
The free energy of a dislocation is made up of
a number of terms:
(i) the core energy (within a radius of about
three lattice planes from the dislocation
(ii) the elastic strain energy outside the core
and extending to the boundaries of the
crystal, and
(iii) the free energy arising from the entropy
In mc the first and second terms increase the
free energy
and are by far the most dominant terms.
Hence dislocations, unlike vacancies, do not
exist in thermal equilibrium.
Increase in strengths and
The relation between
yield stress and grain
size is described
mathematically by the
Hall-Petch equation
where ky is the strengthening coefficient (a constant unique to
each material), σo is a materials constant for the starting stress
for dislocation movement (or the resistance of the lattice to
dislocation motion), d is the grain diameter, and σy is the yield
Grain boundary strengthening
Grain boundary strengthening (or Hall-Petch strengthening) is a
method of strengthening materials by changing their average grain
size. It is based on the observation that grain boundaries impede
dislocation movement and that the number of dislocations within a
grain have an effect on how easily dislocations can traverse grain
boundaries and travel from grain to grain.
So, by changing grain size one can influence dislocation movement and
yield strength.
This is a schematic roughly illustrating the
concept of dislocation pile up and how it effects
the strength of the material. A material with
larger grain size is able to have more dislocation
to pile up leading to a bigger driving force for
dislocations to move from one grain to another.
Thus you will have to apply less force to move a
dislocation from a larger than from a smaller
grain, leading materials with smaller grains to
exhibit higher yield stress.
Hall-Petch strengthening limit
Hall-Petch Strengthening
is limited by the size of
Once the grain size
reaches about 10 nm, grain
boundaries start to slide.
Deformation and fracture of ultra-high-fine materials: (a) Plastic
flow localization; (b) nanockrack nucleation; (c) final failure
Fracture surface of a 30 nm grain size
electrodeposited Ni tensile specimen.
Deformation of nano-metal
from Kumar et al., Acta Materialia, 2003, v.51, 5743 – 5774
How to improve ductility?
NC materials with high
(a) a bimodal single-phase
structure composed
of nanograins and large
grains; and
(b) nano-composite
consisting of nanoscale
grains and dendrite – like
inclusions of the second
(from I.A. Ovid’ko, Rev. Adv. Mater. Sci., 2005,
v.10, 89–104).
Nanostructured solids
Why nanostructured polycrystalline
materials are unstable?
GB consists of several types of extrinsic defects, namely, stationary
dislocations with Burgers vectors normal to a boundary plane, gliding or
tangential dislocations with Burgers vectors tangential to the boundary
plane, and disclinations in triple junctions.
Disclinations and grain boundary dislocations form elastically
distorted layers (zones) near grain boundaries.
High density of defects
-> High energy
Nature -> seek to lower
Grain growth occurs in materials
to reduce the overall energy of the
system by reducing the total grain
boundary energy. Therefore, grain
growth in NC materials is primarily
driven by the excess energy stored
in the grain or interphase
Grain boundaries diffusion
relative increasing of GB diffusion coefficient
Pileups in a grain and a layer of a
nanolayer structure
Nanoscale optical properties
• Bulk gold appears yellow in
• Nanosized gold appears red
in color
– The particles are so small
that electrons are not free to
move about as in bulk gold
– Because this movement is
restricted, the particles react
differently with light
Optical properties are connected with electronic
structure, a change in zone structure leads to a
change in absorption and luminescence spectra.
Visible electromagnetic spectrum
CdSe Semiconducting Quantum Dots
Surface plasmon
Surface plasmon absorption of
spherical nanoparticles and its size
(a) A schematic illustrating the
excitation of the dipole surface
plasmon oscillation. The electric field
of an incoming light wave induces a
polarization of the (free) conduction
electrons with respect to the much
heavier ionic core of a spherical metal
A net charge difference is only felt at
the nanoparticle surfaces, which in turn
acts as a restoring force. In this way a
dipolar oscillation of the electrons is
created with period T.
(b) Optical absorption spectra of 22,
48 and 99nm spherical gold
nanoparticles. The broad absorption
band corresponds to the surface
plasmon resonance (from S. Link,
M.A. El-Sayed Int. Rev. Phys. Chem.
2000, v.19, 409)
Transformation of absorption spectra of sodium from atom
to solid
optical properties
Absorption (fluorescence) spectrum of Na
atom relates to the transition 2S – 2P.
The spectrum of Na3 cluster expands into
the discrete molecular spectrum reflecting
electron excitations and atom oscillations.
Continuous spectrum of Na8 cluster
reflects the processes of dissociations and
defragmentation of cluster on atoms.
Spectrum of nanoparticle reflects
resonance absorption of cluster atoms.
Spectrum of massive film reflects the
interband transitions of electrons in metal.
Optical absorption spectra of sodium:
а) for atom,
b) for cluster Na3,
c) for cluster Na8,
d) for nanoparticle of d<10 nm size (~106 atoms) in
NaCl crystal, e) for thin film of d=10 nm width.
Blue shift
Blue shift refers to a shortening of a transmitted signal's
wavelength, and/or an increase in its frequency. The name comes
from the fact that the shorter-wavelength end of the optical
spectrum is the blue end, hence, when visible light is compacted
in wavelength, it is "shifted towards the blue", or "blue-shifted".
Blue shift phenomenon
is a quantum size effect.
W is a work function, EF is a Fermi energy, HOMO is
the highest occupied molecular orbital, LUMO is the
36lowest unoccupied molecular orbital
Transformation of zone
structure of a solid under
reduction of its size from macroto nano-scale down to a single
atom, showing the increase of
the band gap g ∆E and the blue
shift hω = ∆E for nanoparticles
and nanostructured state of
The properties of MC and NC
materials of the same chemical
In the quantum world, the rules are
Quantum tunneling
A nanoscopic phenomenon in which a particle
violates the principles of classical mechanics by
penetrating a potential barrier or impedance higher
than the kinetic energy of the particle.
Electron tunneling is attained when a particle with lower
energy is able to exist on the other side of an energy
barrier with higher potential energy.
Go through the wall
is the penetration of an electron into a
classically forbidden region.
A barrier, in terms of quantum tunneling, may
be a form of energy state analogous to a "hill"
or incline in classical mechanics, which
classically suggests that passage through or
over such a barrier would be impossible without
sufficient energy.
The principal of quantum tunneling
Electrons exhibit wave behavior and
their position is presented by a wave
(probability) function.
The wave function represents a
finite probability of finding an
electron on the other side of the
potential barrier.
Since the electron does not posses
enough kinetic energy to overcome
the potential barrier, the only way
the electron can appear on the
other side is by tunneling through
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